The local boundedness of the perturbed motions of an imperfect gyroscope in gimbals with dissipative and accelerating forces

Abstract

An unbalanced dynamically symmetrical gyroscope in gimbals with constructive imperfections is considered in a central Newtonian field of forces. It is assumed that there is a moment of forces of viscous friction acting on the axis of rotation of one of the rings of the suspension and an accelerating (electromagnetic) moment applied to the axis of rotation of another ring. The equations of motion have a partial solution for which the basic plane of the frame is perpendicular to the direction from the specified fixed point of the frame to the centre of gravitation, the basic plane of the mantle is parallel to this direction and the rotor rotates with an arbitrary constant angular velocity. The equations of perturbed motions of the reduced system with two degrees of freedom are obtained to within third-order terms at the corresponding position of equilibrium. In the domain of admissible values of the parameters F o the characteristic equation of the system is considered and its coefficients are written down. A domain in F o is specified in which complex conjugate pairs of the eigenvalues have small moduli of the real parts but the absolute values of the second- to fourth-order off-resonance mistuning between the imaginary parts are not small. For an imperfect gyroscope in gimbals with dissipative and accelerating forces the sufficient conditions of the local uniform boundedness of motions perturbed with respect to the specified partial solution are obtained in this domain. The conditions found provide the local uniform boundedness of solutions irrespective of the forms of higher than the third order in the equations of perturbed motions. These conditions are obtained in the form of constraints for the coefficients of the normal form and, finally, for the original parameters of the system and the real and imaginary parts of the eigenvalues. To provide a clear interpretation of the results, special cases when all but two parameters are fixed are analysed. The domains of local uniform boundedness are constructed in the two-dimensional domains F o using a personal computer.